Lemma 2.5 (see [1]} or [2]}) 
Let \(A=\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}\)  be an
\(n\times n\)  positive semidefinite
matrix with diagonal blocks being square, then
\( \prod _{i=1}^n a_{ii}\ge \det A_{11} \det A_{22} \ge \det A. \) 

